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mathematics
calculus with applications

Questions and Answers of Calculus With Applications

In Exercises, determine the interval over which the function is continuous. f(x) = √2x5
In Exercises, determine the interval over which the function is continuous.ƒ(x) = 4x2 - 3x + 7
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = -4x2 + 11x
If the instantaneous rate of change of ƒ(x) with respect to x is positive when x = 1, is f increasing or decreasing there?
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = 6x2 - 4x
Let(a) Graph the function.(b) Determine ƒ(2) and limx→2 ƒ(x). Is ƒ(2) continuous at x = 2?(c) Determine ƒ(3) and limx→3 ƒ(x). Is ƒ(x) continuous at x = 3?(d) Determine ƒ(4) and
Suppose that the total profit in hundreds of dollars from selling x items is given byP(x) = 2x2 - 5x + 6.Find the average rate of change of profit for the following changes in x.(a) 2 to 4 (b) 2
The unemployment rates in the United States for the years 2000–2019 are shown in the graph below. Find the average change per year in unemployment for the following time periods.(a) 2000 to
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = ex
The revenue (in thousands of dollars) from producing x units of an item isR(x) = 10x - 0.002x2.(a) Find the average rate of change of revenue when production is increased from 1000 to 1001 units.(b)
In Exercises, (a) Graph the given function, (b) Find all values of x where the function is discontinuous, and (c) Find the limit from the left and from the right at any values of x
In Exercises, (a) Graph the given function, (b) Find all values of x where the function is discontinuous, and (c) Find the limit from the left and from the right at any values of x
The average price (in cents) per gallon of unleaded gasoline in the United States for the years 2010 to 2019 is shown in the following chart. Find the average rate of change per year in the average
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = ln |x|
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = -2/x
If $1000 is invested in an account that pays 3% compounded annually, the total amount, A(t), in the account after t years isA(t) = 100011.032t.(a) Find the average rate of change per year of the
Find the x-values where the following do not have derivatives. VA 2+ 0 -2 +2 X
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = 6/x
If $1000 is invested in an account that pays 3% compounded continuously, the total amount, A(t), in the account after t years isA(t) = 1000e0.03t.(a) Find the average rate of change per year of the
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim (2x² - 4x + 5) x-3
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = √x
The graph shows the money remaining in the Medicare Trust Fund at the end of the fiscal year.Find the approximate average rate of change in the trust fund for each time period.(a) From 2015 to
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = -3 √x
In Exercises, (a) Graph the given function, (b) Find all values of x where the function is discontinuous
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim (√x - x) x→8
In Exercises, (a) Graph the given function, (b) Find all values of x where the function is discontinuous, and (c) Find the limit from the left and from the right at any values of x
The graph shows the temperature T in degrees Fahrenheit as a function of the altitude h in feet when an inversion layer is over Southern California. (An inversion layer is formed when air at a higher
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x-2 9- X-zX x x + 2
The following chart shows how the percentage of eighth graders, tenth graders, and twelfth graders who have used marijuana in their lifetime has varied in recent years.(a) Find the average annual
In Exercises, (a) Use a graphing calculator to tell where the rational function P(x)/Q(x) is discontinuous, and (b) Verify your answer from part (a) by using the graphing calculator to plot
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x-5 x². X - 3x x-5 10
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x-0 1/(x + 3) - 1/3 X
A car is moving along a straight test track. The position in feet of the car, s(t), at various times t is measured, with the following results.Find and interpret the average velocities for the
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 9-x^ lim x→36 x — 36
In Exercises, find the value of the constant k that makes the function continuous. f(x) = if x ≤ 2 Skx² [√x + k if x > 2
The future size of the world population depends on how soon it reaches replacement-level fertility, the point at which each woman bears on average about 2.1 children. The graph shows projections for
Find the x-values where the following do not have derivatives. + -6 + y 8 0 6 X
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x-2x² x + 2 - x - 6
Find the x-values where the following do not have derivatives. -3 1 1 I 1 1 1 f(x) + 0 -3- X
Find the x-values where the following do not have derivatives. ا f(x) 3 نرا 10 2 | | | | ا | X
In Exercises, find the value of the constant k that makes the function continuous. g(x) = [x³ + k_ifx ≤ 3 kx 5 if x > 3 -
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. x - 3 5x +4 lim x→-3x2 +
In Exercises, find the value of the constant k that makes the function continuous. h(x) = 3x² + 2x x + 2 3x + k 8 if x = -2 if x = -2
In Exercises, find the value of the constant k that makes the function continuous. g(x) = 2x² - x - x-3 kx - 1 15 if x # 3 if x = 3
The number of cases of measles in the United States for recent years is shown in the graph at the top of the next column. Find the average rate of change per year for each of the following time
The crown length (as shown below) of first molars in fetuses is related to the postconception age of the tooth asL(t) = -0.01t2 + 0.788t - 7.048,where L(t) is the crown length, in millimeters, of the
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 3 X E-X 6-zx lim
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. x² lim x 2 x + 4 2
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x→1 5x² - 7x + 2 x² - 1
Find the x-values where the following do not have derivatives.Suppose g(x) = 3√x . Use the graph of g(x) to find g102.
In Exercises, (a) Use a graphing calculator to tell where the rational function P(x)/Q(x) is discontinuous, and (b) Verify your answer from part (a) by using the graphing calculator to plot
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. x² - 9 2 X lim x→−3 x² + x − 6
Explain in your own words what the Intermediate Value Theorem says and why it seems plausible.
The metabolic rate of a person who has just eaten a meal tends to go up and then, after some time has passed, returns to a resting metabolic rate. This phenomenon is known as the thermic effect of
By considering, but not calculating, the slope of the tangent line, give the derivative of the following.(a) ƒ(x) = 5 (b) ƒ(x) = x(c) ƒ(x) = -x (d) The line x = 3(e) The line y = mx + b
Explain why limx→2 (3x2 + 8x) can be evaluated by substituting x = 2.
The following graph shows how immigration (in thousands) to the United States has varied from 1910 to 2010. The following chart shows how the percentage of eighth graders, tenth graders, and twelfth
If the rate of change of ƒ(x) is zero when x = a, what can be said about the tangent line to the graph of ƒ(x) at x = a?
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x-0 -1/(x + 2) + 1/2
The graph shows the profit from the daily production of x thousand kilograms of an industrial chemical. Use the graph to find the following limits.(a) limx→6 P(x) (b) limx→10-
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. √x - 5 lim x 25 x 25
In Exercises, find the derivative of the function at the given point.(a) Approximate the definition of the derivative with small values of h.(b) Use a graphing calculator to zoom in on the function
The cost to transport a mobile home depends on the distance, x, in miles that the home is moved. Let C(x) represent the cost to move a mobile home x miles. One firm charges a flat fee of $100, plus
Consider the example at the beginning of this section regarding the car traveling from San Francisco.(a) Estimate the instantaneous velocity at 1 hour. Assume that the velocity changes at a steady
In Exercises, find the derivative of the function at the given point.(a) Approximate the definition of the derivative with small values of h.(b) Use a graphing calculator to zoom in on the function
The distance in feet of a particle from some fixed point is given bys(t) = t2 + 5t + 2,where t is time measured in seconds. Find the average velocity of the particle over the following intervals.(a)
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim h→0 (x + h)² - x² h
In Exercises, find the derivative of the function at the given point.(a) Approximate the definition of the derivative with small values of h.(b) Use a graphing calculator to zoom in on the function
For each function in Column A, graph [ƒ(x + h) - ƒ(x)4/h for a small value of h on the window [-2, 2] by [-2, 8]. Then graph each function in Column B on the same window. Compare the first set of
A company charges $1.25 per lb for a certain fertilizer on all orders 100 lb or less, and $1 per lb for orders over 100 lb. Let F(x) represent the cost for buying x lb of the fertilizer. Find the
Recently, a car rental firm charged $36 per day or portion of a day to rent a car for a period of 1 to 5 days. Days 6 and 7 were then free, while the charge for days 8 through 12 was again $36 per
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim h→0 (x + h)³ = x³ h
In 2020, it cost $1.00 to send a large envelope within the United States for the first ounce, and $0.20 for each additional ounce, or fraction thereof, up to 13 ounces. Let C(x) be the cost to mail x
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim (x² + 4x 3) - X-∞
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim (x³ + 2x + 8) X--∞
A woman’s weight naturally increases during the course of a pregnancy. When she delivers, her weight immediately decreases by the approximate weight of the child. Suppose that a 120-lb woman gains
Suppose the demand for a certain item is given by D(p) = -2p2 - 4p + 300, where p represents the price of the item in dollars.(a) Find the rate of change of demand with respect to price.(b) Find and
Researchers at Iowa State University and the University of Arkansas have developed a piecewise function that can be used to estimate the body weight (in grams) of a male broiler during the first 56
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x→ 7x 3x 1
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 8x + 2 lim x--∞ 4x - 5
The revenue in dollars generated from the sale of x picnic tables is given by(a) Find the marginal revenue.(b) Estimate the revenue from the sale of the 1001st table by finding R′(1000).(c)
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 3x² + 2x lim x→∞ 2x²² - 2x + 1 -00
The average cost per unit, C̅(x), in dollars, to produce x units of yogurt is given by(a) Find the marginal average cost.(b) Find and interpret the marginal cost at a production level of 50 units of
The profit (in thousands of dollars) from the expenditure of x thousand dollars on advertising is given by P(x) = 1000 + 32x - 2x2. Find the marginal profit at the following expenditures. In
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim 00-X x² + 2x - 5 3x² + 2
Suppose a cost-benefit model is given bywhere ƒ(x) is the cost (in thousands of dollars) of removing x percent of a certain pollutant from an area stream.(a) Find the rate of change of the cost with
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 3x³ + 2x - 1 3x³ - 2 lim xx2x4
The cost in dollars of producing x tacos is C(x) = -0.00375x2 + 1.5x + 1000, for 0 ≤ x ≤ 180.(a) Find the marginal cost.(b) Find and interpret the marginal cost at a production level of 100
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 2.x² 1 1 lim x3x² + 2
Many birds, such as cockatiels or the Arctic terns shown below, have flight muscles whose expenditure of power varies with the flight speed in a manner similar to the graph shown on the next page.
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 2x³ - x - 3 lim x→∞0 6x² 2 - x - 1 I
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 2x² - 7x4 lim x 9x2 + 5x - 6
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. -5x³ 4x² + 8 lim x 6x² + 3x + 2
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x →∞ x4 - x³ - 3x 7x² + 9 2
The graph below shows the temperature one Christmas day of a Heat-Kit Bakeoven, a wood-burning oven for baking. The oven was lit at 8:30 a.m. Let T(x) be the temperature x hours after 8:30 a.m.(a)
In one research study, the population of a certain shellfish in an area at time t was closely approximated by the following graph. Estimate and interpret the derivative at each of the marked points.
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. Let g(x) = So 1/1/x² – 3 - if x = -2 -2 3 if x Find lim g(x). x--2
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. Let f(x) = = [x³ + 2 if x # −1 5 if x = -1 Find lim f(x). x-1
The graph shows how the velocity of a baseball that was traveling at 90 miles per hour when it was hit by a Major League baseball player varies with respect to the weight of the bat.(a) Estimate and
The graph shows how the velocity of a baseball that was traveling at 40 miles per hour when it was hit by a Little League baseball player varies with respect to the weight of the bat.(a) Estimate and

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